Crossed Products and Cleft Extensions for Coquasi-hopf Algebras

The notion of crossed product with a coquasi-Hopf algebra H is introduced and studied. The result of such a crossed product is an algebra in the monoidal category of right H-comodules. We give necessary and sufficient conditions for two crossed products to be equivalent. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed product. Second, the category of coquasi-Hopf modules is trivial, namely equivalent to the category of modules over the starting associative algebra. In connection the crossed product, we recall from [1] the notion of a cleft extension over a coquasi-Hopf algebra. A Morita context of Hom spaces is constructed in order to explain these extensions, which are shown to be equivalent with crossed product with invertible cocycle.